Classification of k-nets

A finite k-net of order n is an incidence structure consisting of k >= 3 pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the k classes. Deleting a line class from a k-net, with k >= 4, gives a derived (k 1)-net of the same order. Finite k-nets embedded in a projective plane PG(2, K) coordinatized by a field K of characteristic 0 only exist for k = 3, 4, see Korchmaros et al. (2014). In this paper, we investigate 3-nets embedded in PG(2, K) whose line classes are in perspective position with an axis r, that is, every point on the line r incident with a line of the net is incident with exactly one line from each class. The problem of determining all such 3-nets remains open whereas we obtain a complete classification for those coordinatizable by a group. As a corollary, the (unique) 4-net of order 3 embedded in PG(2, K) turns out to be the only 4-net embedded in PG(2, K) with a derived 3-net which can be coordinatized by a group. Our results hold true in positive characteristic under the hypothesis that the order of the k-net considered is smaller than the characteristic of K. (C) 2015 Elsevier Ltd. All rights reserved

3-Nets realizing a group in a projective plane

In a projective plane PG(2,K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614–1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672–688, 2008), comprises 3-nets realizing dihedral groups.We prove that there is no further infinite family. Urzúa’s 3-nets (Adv. Geom. 10:287–310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p >0 apart from three possible exceptions Alt4, Sym4, and Alt5. Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069–1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469–488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672–688, 2008; Yuzvinsky in Compos. Math. 140:1614–1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641–1648, 2009

Classification of Web Forum Entries

Diplomová práce se zabývá klasifikací textu v prostředí internetu. Jsou v ní popsány dostupné moderní metody pro klasifikaci a následné rozdělení textových příspěvků. Součástí diplomové práce je implementace Bayesova naivního algoritmu a klasifikátoru s využitím neuronových sítí. Vybrané metody jsou zde porovnány vzhledem k jejich chybovosti či jiným klasifikačním vlastnostem.This thesis is dealing text ranking on the internet background. There are described available methods for classification and splitting of the text reports. The part of this thesis is implementation of Bayes naive algorithm and classifier using neuron nets. Selected methods are compared considering their error rate or other ranking features.

Classification of Local Conformal Nets. Case c < 1

We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of A-D_{2n}-E_{6,8} Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c<1 with certain coset nets. Then, by using the classification of modular invariants for the minimal models by Cappelli-Itzykson-Zuber and the method of alpha-induction in subfactor theory, we classify all local irreducible extensions of the Virasoro nets for c<1 and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the literature.Comment: 30 pages, LaTeX2